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ERNEST ORLANDO LAWRENCE BERKELEY N-ATICJNAL LABORATORY
Fourier's Heat Conduction Equation: History, Influence, and Connections
T.N. Narasimhan
Earth Sciences Division
May 1998 Invited article to appear in Reviews of Geophysics
This document was prepared as an account of work sponsored by the United States Government. While this document is believed to contain correct information, neither the United States Government nor any agency thereof, nor the Regents of the University of California, nor any of their employees, makes any warranty, express or implied, or assumes any legal responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by its trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof, or the Regents of the University of California. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof or the Regents of the University of California.
Fourier's Heat Conduction Equation: History, Influence, and Connections
T .N. Narasimhan
Department of Materials Science and Mineral Engineering Department of Environmental Science, Policy and Management
University of California, Berkeley
and
Earth Sciences Division Ernest Orlando Lawrence Berkeley National Laboratory
University of California Berkeley, CA 94720
May 1998
LBNL-41798
This work was supported in part by the Director, Office of Energy Research, Office of Basic Energy Sciences, of the U.S. Department of Energy under Contract No. DE-AC03-76SF00098.
FOURIER'S HEAT CONDUCTION EQUATION: HISTORY, INFLUENCE, AND CONNECTIONS
T.N. NARASIMHAN
Department of Materials Science and Mineral Engineering Department of Environmental Science, Policy and Management
Earth Sciences Division, Ernest Orlando Lawrence Berkeley National Laboratory . 467 Evans H~ University of California at Berkeley
Berkeley, CA 94720-1760
ABSTRACT
The equation describing the conduction of heat in solids has, over the past two centuries, proved to be a very powerful tool for analyzing the dynamic motion of heat as well as for solving an enormous array of diffusion-type problems in physical sciences, biological sciences, earth sciences, and social sciences. This equation was formulated at the beginning of the nineteenth century by one of the most gifted scholars of modem science, Joseph Fourier of France. A study of the historical context in which Fourier made his remarkable contribution and the subsequent impact his work has
. had in the development of modem science is as fascinating as it is educational This paper is an attempt to present a picture of how certain ideas initially led to the development of the heat equation by Fourier and how, subsequently, Fourier's work directly influenced and inspired others to use the heat diffusion model to describe other dynamic physical systems. Conversely, others concerned with the study of random processes, found that the equations governing such random processes reduced, in the limit, to Fourier's equation of heat diffusion. In the process of developing the flow of ideas, the paper also presents, to the extent possible, an account of the history and personalities involved.
Page 1
INTRODUCTION
The equation describing the conduction of heat in solids occupies a unique position in modern
mathematical physics. In addition to lying at the core of analyzing problems involving the dynamic
transfer of heat in physical systems, the conceptual-mathematical structure of the heat conduction
equation (also known as heat diffusion equation) has inspired the mathematical formulation of many
other physical processes in terms of diffusion. As a consequence, the mathematics of diffusion has
helped the transfer of knowledge relating to problem solving among diverse, seemingly unconnected
disciplines. The transient process of heat conduction is described by a partial differential equation,
familiarly referred to as a parabolic equation, which was first formulated by Jean Baptiste Joseph
Fourier (1768-1830) in 1807 and presented as a manuscript to the Institut de France. At the time this
manuscript was prepared, thennodynamics, potential theory, and differential equations were all in the
initial stages of their formulation. Combining remarkable gifts in pure mathematics and insights into
observational physics, Fourier opened up new areas of investigation in mathematical physics with his
masterpiece, Theorie de Ia propagation de Ia Chaleur dans les So/ides (Fourier, 1807).
Fourier's work was subjected to review by some of the most distinguished scientists of the
time. However, it was not accepted as readily as one might have expected. It would be another
fifteen years before this major contribution would be accessible to the general scientific community
through publication of his classic monograph, Theorie analytique de Ia Chaleur (Fourier, 1822).
Soon after this publication, the power and significance of Fourier's work was recognized outside of
France. Fourier's method began to be applied to analyze problems in many fields besides heat
transfer: electricity, chemical diffusion, fluids in porous media, genetics, and economics. It also
inspired a great deal of research into the theory of differential equations. Nearly two centuries later,
the heat conduction equation continues to constitute the conceptual foundation on which rest the
analysis of many physical, biological, and social systems.
A study of the conditions that led to the articulation of the heat conduction equation and the
reasons why that equation has had such a major influence on scientific thought over nearly two
centuries is in itself very rewarding. At the same tirre, an examination of how the work was received
and accepted by Fourier's peers and successors gives us a fascinating glimpse into the culture of
science, especially as it prevailed during the nineteenth century in Europe. The present work has been
Page2
motivated both by the educational and historical importance of Fourier's work. Accordingly, the
purpose of this·paper is to explore how the framework of the heat conduction equation has come to
help us understand an impressive array of seemingly disconnected natural processes. In so doing, the
purpose also is to gain historical insights into the manner in which scientific ideas develop.
The paper starts with pertinent scientific developments during the eighteenth century that set
the stage for Fourier's work on heat conduction. Following this, details are presented of Fourier
himself and his contribution, especially the 1807 manuscript. Fourier's influence has occurred along
two lines. Experimentalists in electricity, chemical diffusion and fluid flow in porous materials
directly derived benefit by interpreting their experiments by analogy with the heat conduction
phenomenon. Researchers in other fields such as statistical mechanics and probability theory
indirectly established connections with the heat conduction equation by recognizing the similarities
between the mathematical behavior of their systems and mathematical solutions of the heat
conduction equation. These direct and indirect influences of Fourier's work are described next. The
paper concludes with some reflections . on the scientific atmosphere during the early nineteenth
century, a comparison of the different facets of diffusion and a look beyond Fourier's solution
strategy. A chronology of the important developments is presented in Table 1.
DEVELOPMENTS LEADING UP TO FOURIER
Before·we describe the scientific developments of the eighteenth century that set the stage for
Fourier's contribution, it is pertinent to briefly state the nature and content of the heat conduction
process. The transient heat conduction phenomenon as embodied in Fourier's partial differential
equation pertains to the conductive transport and storage of heat in a solid body. The body itself, of
finite shape and size (e.g., a rod, a cylindrical annulus, a sphere, a cube), communicates with the
external world by exchanging heat across its boundary. Within the solid body, heat manifests itself
in the form of temperature, which can be measured accurately. Under these conditions, Fourier's
differential equation mathematically describes the rate at which temperature is changing at any
location in the interior of the solid as a function of time. Physically, the equation describes the
conservation of heat energy per unit volume over an infinitesimally small volume of the solid centered
at the point ofinterest. Crucial to such conservation of heat is the recognition that 'heat continuously
Page 3
TABLE 1: A CHRONOLOGY OF SIGNIFICANT CONTRIBUTIONS ON DIFFUSION
Fahrenheit Abbe Nollet Bernoulli Black Crawford Lavoisier and Laplace Laplace Biot Fourier Fourier Olun Dutrochet Green Graham Thompson Poiseuille Graham Fick Darcy Dupuit Maxwell Pfeffer Edgeworth Forchheimer van't Hoff Nernst Lord Rayleigh Roberts-Austen Bacheller Einstein Pearson Pearson Buckingham Langevin Gardner Fisher Terzaghi Richards Fermi Kolrnogorov Chandrasekhar Taylor Samuelson Merton
Mercury thennometer and standardized temperature scale Observation of osmosis across animal membrane Use of trigonometric series for solving differential equation Recognition of latent heat and specific heat Correlation between respiration of animals and their body heat First calorimeter; measurement of heat capacity, latent heat Formulation of Laplace operator Heat conduction among discontinuous bodies Partial differential equation for transient heat conduction in solids Theorie Analytique de Ia Chaleur Law governing current flow in electrical conductors Discovery of endosmosis and exosmosis Formal definition of a potential Law governing diffusion of gases Similarities between equations of heat diffusion and electrostatics Experimental studies on water flow through capillaries Experimental studies on diffusion in liquids Fourier's model applied to molecular diffusion Law governing flow of water in porous media Potential theory applied to flow of groundwater in geological basins Diffusion equation for gases derived from dynamical theory Investigations on osmosis in biological and inorganic membranes Law of error and Fourier equation Flownets for solving seepage problems using potential theory Theory of osmotic pressure by analogy with gas laws Interpretation of Fick' s law in terms of forces and resistances Random mixing of sound waves as a diffusion process Experimental measurement of solid diffusion Option pricing and diffusion of probability . Brownian motion and diffusion equation The notion of random walk Random migration of animals as a diffusion problem Diffusion of multiple fluid phases in soils Framework for stochastic differential equation Measurement of potential in a multi-fluid-phase porous medium Inheritance of genes as a diffusion problem Transient seepage in deformable clays as analogous to heat diffusion Non-linear diffusion of moisture in soils Neutron diffusion in graphite as analogous to heat diffusion Traveling wave solution to non-linear diffusion Generalization of Lord Rayleigh's stochastic differential equations Advective dispersion as a diffusion process Warrant pricing and diffusion equation Stochastic calculus and theory of option pricing
moves across the surfaces bounding the infinitesimal element as ·dictated by the variation of
temperature from place to place within the solid and that the change in temperature at a point reflects
the change in the quantity of heat stored in the vicinity of the point.
It is clear from the above that the notions of temperature, quantity of heat, the relation
between quantity of heat and temperature, and the notion of transport of heat are fundamental to the
formulation of Fourier's equation. It is important to recognize here that these basic notions were still
evolving at the end of the eighteenth century. Therefore, it is appropriate for us to begin by
familiarizing ourselves with the evolution of these notions during the eighteenth century.
Since heat can be readily observed and measured only in terms of temperature, the
development of a reliable thennOireter capable of giving repeatable measurements was critical to the
growth of the science of heat Gabriel Daniel Fahrenheit (1686-1736), a German instrument maker
and physicist, perfected the closed-tube mercury thermometer in 1714 and conunercially produced
them by 1717 (Middleton, 1966). By 1724 he had established what we now know as the Fahrenheit
scale with the melting of ice at 32° and the boiling of water at 212°. Fahrenheit succeeded in
calibrating his instruments carefully so that measurements were accurate and reliable.
The next developments of interest were qualitative and conceptual, and of great importance.
Joseph Black (1728-1799), a pioneer in quantitative chemistry, was known for his lectures in
chemistry at Glasgow and was also a practicing physician. Around 1760 he noticed that when ice
melts it takes in heat without changing temperature. This observation led him to propose the term
''latent heat" to denote the heat taken up by water as it changes its state from solid to liquid. He also
noticed that equal masses of different substances needed different amounts of heat to raise their
temperatures by the same amount. He coined the term "specific heat" to denote this type of heat.
Although Black is said to have constructed an ice calorimeter, he never published his results. The
precise measurement of latent heat and specific heat was left to Lavoisier and Laplace, some twenty
years later. Another important development was the appearance of the book Experiments and
Observations of Animal Heat, and the Inflammation of Combustible Bodies by Adair Crawford
(1748-1795) in 1779. In this work, Crawford proposed that oxygen was involved in the generation
of heat by animals during respiration and went on to discuss a method of measuring specific heat by
a rrethod of mixtures (Guerlac, 1982). Crawford's idea of measuring specific heat by the method of
Page4
mixtures would soon have a significant influence on Lavoisier and Laplace, although he himself was
unable to measure these quantities accurately.
In the wake of the contributions of Black and Crawford, what must be considered as one of
the most important papers of modem chemistry and thermodynamics appeared in 1783. This was the
paper entitled M emoire sur Ia Chaleur coauthored by Antoine Laurent Lavoisier ( 17 43-1794 ), the
central figure of the revolution in chemistry of the later eighteenth century, and Pierre Simon Laplace
(1749-1827), one of the most influential mathematician and theoretical physicists of modern science.
This paper provided detailed descriptions of an ice caloriireter with which they measured, for the first
time, the latent heat of melting of ice and the specific heats of different materials. All the
measurements were made relative to water, the chosen reference. They also showed experimentally
that heat is released during respiration of animals, by placing a guinea pig within the calorimeter for
several hours and measuring the quantity of ice melted. In a related set of experiments, they also
demonstrated quantitatively that the process of respiration, in which oxygen is combined with carbon
in the animal's body, is in fact combustion, resulting in the release of heat. During the late nineteenth
century when this work was done, the nature of heat was still a matter of debate among scientists.
Some believed that heat was a fluid diffused within the body (referred to as "caloric") while others
believed that heat was a manifestation of vibrations or motions of molecules. Although Lavoisier
and Laplace preferred the latter concept, they interpreted and presented their results in such a way
that the experiments stood by themselves, independent of any hypothesis concerning the nature of
heat In so far as Fourier's heat conduction equation is concerned, the significance of the Lavoisier
Laplace work is that it provided the notion of specific heat, which is fundamental to the understanding
of time-dependent changes of temperature. Nonetheless, the significance of the work far transcends
Fourier's equation. By experimentally quantifying latent heat and heats of reactions, the Lavoisier
Laplace work constitutes an essential component of the foundations of thermodynamics.
We now consider the process of transfer of heat in solids, that is, the process of heat
conduction. The best known work in this regard is that of Jean BaptiSte Biot (1774-1862), a versatile
scientist who made important contributions in magnetism, optics, and celestial mechanics. Biot
(1804) addressed the problem of heat conduction in a thin bar heated at one end (Grattan-Guinness,
1972). In the bar, heat was not only conducted along the length but it was also lost to the exterior
PageS
atmosphere transverse to the direction of conduction. His starting point to analyze this problem was
Newton's law of cooling, according to which the rate at which a body loses heat to its surroundings
is proportional to the difference in temperature between the bar and the exterior atmosphere. Biot,
who was a student of Laplace's rrechanistic schooL believed in the philosophy of action at a distance
between bodies. Accordingly, the temperature at a point in the heated rod was perceived to be
influenced by all the points in the vicinity of the point. Essentially then, the mathematical problem
of heat conduction carne to be considered as one of a class of many-body problems. As pointed out
by Grattan-Guinness (1972), Biot's idealization of action at a distance involved only the difference
in temperature between points and did not involve the distance between the points. As a
consequence, Biot's approach did not involve a temperature gradient, so necessary to the formulation
of the differential equation. However, Biot did articulate the underlying concepts clearly by stating
that when the heat content of the bar changes at each instant, the net accumulation of heat at a point
causes a change in temperature. Biot also asserted that he experimentally found ~~-~ton's ~w
concerning the loss of heat to be rigorous. It is not quite clear how Biot chose to work on the heat
conduction problem A footnote in his paper refers to earlier experiments of Count Rumford, but no
other details are given.
Apart :(rom these foundational developments pertaining to heat, two other major topics of the
eighteenth century are pertinent to Fourier's work: potential theory and differential equations. The
theory of potentials arises in many branches of science such as electrostatics, magnetostatics,
irrotational movement of perfect fluids, and so on. Potential theory involves problems describable
in terms of a partial differential equation in which the dependent variable is the appropriate potential
"(defined as energy per unit mass, charge, etc.) and the sum of the second spatial derivatives of the
potential in three principal directions is equal to zero. This equation was first formulated by Laplace
in 1789, although the term potential would be coined later by George Green (1793-1841), a self
educated mathematician, in a classic essay on electricity and magnetism (Green, 1828). Laplace
formulated the equation in the context of the problem of the stability of Saturn's rings (Laplace,
1789). The mathematical operator denoting the sum of the second spatial derivatives of the potential
is therefore known as the Laplacian and the equation itself is known as Laplace's equation in his
honor.
Page6
The eighteenth century also saw very active developments in the theory of ordinary and partial
differential equations through the contributions of Daniel Bernoulli (1700-1782), Jean le Ronda
d'Alembert (1717-1783), Leonhard Euler (1707-1783), John-Louis Lagrange (1736-1813), and
others. For the partial differential equation describing a vibrating string, Daniel Bernoulli had
suggested, on physical grounds, a solution in terms of trigonometric series. Similar usage of
trigonometric series was also made a little later by Euler and Lagrange. Yet, d' Alembert, Euler, and
Lagrange were not particularly satisfied with the trigonometric series. Their concerns were purely
mathematical in nature, consisting of issues of convergence and algebraic periodicity of such series
(Grattan-Guinness, 1972).
It is pertinent here to dwell a little on the atmosphere of scientific philosophy that existed in
Europe at the turn of the nineteenth century. Two views of the physical world prevailed at the time:
the mechanistic school of Isaac Newton (1642-1727) and the dynamic school of Gottfried Wilhelm
Leibniz (1646-1716). During the eighteenth and nineteenth centuries, a number of the most gifted
thinkers from France were fully committed to the mechanistic view and devoted their efforts to
describing the physical world with grater and greater detail in terms of Newton's laws. At the same
time, his contemporary Leibniz also had a major influence on the development of scientific thought.
At the foundation of physics were the notions of force, momentum, work and action. Although these
notions are all related, Newton and Leibniz pursued two parallel, but distinct avenues to
understanding the physical world. Newton's approach was based on the premise that by knowing
the forces and momenta at every point or particle, one could completely describe a physical system
Leibniz, on the other hand pursued the approach of understanding the total system in terms of work
and action. One of the leading figures of Newton's mechanistic school was Laplace. Laplace, in tum,
had many ardent followers, including Biot and Poisson. Among those who followed Leibniz's
philosophy were Lagrange, Euler and Hamilton. Although, ultimately, both approaches proved
equivalent, the mathematics associated with them are very different. While the mechanistic school
relied on the use of vector fields to describe the physical system, the dynamic school of Leibniz could,
remarkably, realize the same results through the use of energy and action, which are scalar quantities.
Additionally, the thinking of the mathematical physicists of the late eighteenth century was also
influenced by their intense interest in celestial mechanics, a field which had greatly captivated Galilee,
Page7
Newton and Kepler.
It was under these circumstances that observational data on heat, electricity, chemical
reactions, and pbysiology of animals were being collected and great efforts were being made to
rationally understand them in terms of force, momentum, energy and work. As should be expected,
opposing views were pursued and tested before concepts and ideas could evolve into forms that we
now take for granted. When Fourier commenced his work on heat conduction at the turn of the
nineteenth century, the nature of heat was still unresolved. Those of the mechanistic school, including
Biot, believed that heat was a permeating fluid. On the other hand, those of the dynamic school
believed that heat was essentially motion, rapid molecular vibrations. Those of the mechanistic school
also believed that a cogent theory of heat should be rigorously ~uilt from a detailed description of
motion at the level of individual particles. This approach, it appears, governed the work of Biot
(1804) and his use of action at a distance.
FOURIER'S CONTRffiUTION
As we have seen, the science of heat, the theory of potentials and the theory of differential
equations were all in their early stages of development by the time Fourier started his work on heat
conduction. Opinions were still divided about the nature of heat: whether it was an all-pervading fluid
or it was related to molecular motion. However, heat conduction due to temperature differences and
heat storage and the associated specific heat of materials had been experimentally established.
Potential theory had already been formu1ated, and both the Laplace equation and the Laplace operator
were well established. Frilally, the representation of dynamic problems in continuous media with the
help of partial differential equations (e.g., the problem of a vibrating string) and their solution with
the help of trigonometric series were also known. This is the context in which Fourier began working
on the transient heat conduction problem
Fourier's life and contributions are so unusual that a brief sketch of his career and the
conditions under which he worked are very worthwhile. For a comprehensive account, the reader
is referred to Grattan-Guinness (1972). Joseph Fourier was born in 1768 in Auxerre in Burgundy,
now the capital of Yonne Departement in central France. In 1789, about the time his mathematical
talents began to blossom, the French Revolution intervened. In his native Auxeire he was socially
Page8
and politically active, being a forceful orator. His outspoken criticism of corruption almost took him
to the guillotine in 1794; he was saved mainly by the public outcry in the town and a deputation of
local people on ~ behalf. Following this he taught mathematics for a few years at the Ecole
Polytechnique in Paris. In 1798, Napoleon Bonaparte (1769-1821) was leading an expedition to
Egypt and Fourier was made Secretaire Perpetuel of the newly formed Institut d 'Egypte. In Egypt
he held many important administrative and judicial positions, and in 1799 was made leader of a .
scientific expedition investigating monuments and inscriptions in Upper Egypt. In November 1801
Fourier returned to France upon the withdrawal of French forces from Egypt. However, his hopes
for resuming his teaching duties at Ecole Polytechnique were ended when Bonaparte made him
Prefect of the Department oflsere near the Italian border, with its capital at Grenoble.
During his tenure as Prefect, a demanding job that lasted many years, Fourier embarked on
two very different major scholarly efforts. On the one hand he started his leadership role on a multi
volume work on Egypt, which would later form the foundation for the science of Egyptology. On
the other, he began working on the problem of heat diffusion. It appears that Fourier started work
on heat conduction sometime between 1802 and 1804 (Grattan-Guinness, 1972), probably for no
other reason than that he saw it as one of the unsolved problems of his time. Between 1802 and
1807, he conducted his researches into Egyptology· and heat diffusion whenever he could find spare
time from his prefectural duties.
Just as Biot before him, Fourier initially formula~ the heat conduction problem as an n-body
problem, stemming from the Laplacian philosophy of action at a distance. During this early
investigations, he was aware ofBiot's work, having received a copy ofBiot's paper from the author
himself. For sorre reason that is not quite clear, Fourier abandoned the action at a distance approach
around 1804 and made a bold departure from convention, which eventually led to his masterpiece,
the transient heat conduction equation.
Essentially, what Fourier did was to move away from discontinuous bodies and towards
continuous bodies. Instead of starting with the basic equations of action at distance, Fourier took an
empirical, observational approach to idealize how matter behaved macroscopically. In this way he
also avoided discussion of the nature of heat. His observational approach (like that of Lavoisier and
Laplace before him), was independent of what the fundamental nature of heat was. Rather than
Page9
assuming that the behavior of temperature at a point was influenced by all points in its vicinity,
Fourier asswred that the temperature in an infinitesimal lamina or element was dependent only on the
conditions at the lamina or element ii:mrediately upstream and downstream of it. He thus formulated
the heat diffusion problem in a continuum.
In fomrulating heat conduction in terms of a partial differential equation and developing the
methods for solving the equation, Fourier initiated many innovations. He visualized the problem in
tenns of three components: heat transport in space, heat storage within a small element of the solid,
and boundary conditions. The differential equation itself pertained only to the interior of the flow
domain. The interaction of the interior with the exterior across the boundary was handled in terms
of''boundary conditions", conditions assumed to be known a priori. The parabolic equation devised
by Fourier was a linear equation in which the parameters, conductivity and capacitance were
independent of time or temperature. This attribute of linearity enabled Fourier to draw upon the
powerful concept of superposition to combine many particular solutions and thereby create general
solutions (Grattan-Guinness, 1972). The superposition artifice offered such promise in solving
problems that mathematicians who followed Fourier resorted to linearizing differential equations so
as to facilitate their subsequent solution.
Perhaps the most powerful and most daunting aspect of Fourier's work was the method of
solution. Fourier was clearly aware of the earlier work of Bernoulli, Euler, and Lagrange relating to
solutions in the form of trigonometric series. He was also aware that Euler, D'Alembert and
Lagrange viewed trigonometric series with great suspicion. Their opposition to the trigonometric
series stetnrred from reasons of pure mathematics: convergence and algebraic periodicity. Lagrange,
in fact had a particular preference for solutions expressed in the form of Taylor series (Grattan
Guinness, 1972). Yet, Fourier, who was addressing a well-defined physical problem with physically
realistic solutions, did not allow himself to be held back by the concerns of his illustrious
predecessors. He boldly applied the method of separation of variables and generated solutions in
tenns of infinite trigonometric series. Later, he would also generate solutions in the form of integrals
that would come to be known as Fourier integrals. In the last part of his 1807 work, Fourier also
presenteo some results pertaining to heat conduction in a cylindrical annulus, a sphere and a cube.
Fourier submitted his manuscript to the French Academy in December 1807. As was the
Page 10
practice, the secretary of the Academy appointed a conunittee of reviewers consisting of four of the
most renowned mathematicians of the t::ilre, Laplace, Lagrange, Monge, and Lacroix. The manuscript
was not well received, particularly by Laplace and Lagrange, for the mathematical reasons alluded
to above. Although Laplace would later become sympathetic to Fourier's method, Lagrange would
never change his mind. Because of the lack of approval by his peers, the possible publication of
Fourier's wor~ by the French Academy was getting delayed indefinitely. In the end, Fourier took it
upon himself to expand the work and publish it on his own in 1822 under the title, Theorie
Analytique de Ia Chaleur, which is now an avowed classic.
THE HEAT CONDUCTION EQUATION
It is appropriate to introduce here the transient heat conduction equation of Fourier. In
modem notation, this parabolic partial differential equation may be written as,
(1) V·KVT aT = c-at
where K is thermal conductivity, T is temperature, c is specific heat capacity of the solid per unit
volume and t is time. The dependent variable T is a scalar potential while thermal conductivity and
specific heat capacity are empirical parameters. Physically, the equation expresses the conservation
of heat per unit volume over an infinitesimally small volume lying in the interior of the flow domain.
The exchange of heat with the external world is to be taken into account with the help of either
temperature or thermal fluxes prescribed on the boundary. Also, it is assumed that the distribution
of temperature over the domain is known at the initial time t = 0.
For the particular case when the temperature over the flow domain does not change with time
and is steady, (I) reduces to the Laplace equation,
(2) V·KVT = 0 .
Page 11
The left hand sides of (1) and (2) are the Laplacian or the Laplace operator referred to earlier.
It is appropriate here to pay attention to the physical parameters of the above equations. The
thermal conductivity K is a constant of proportionality, which relates the quantity of heat crossing
a unit surface area in unit time to the spatial gradient of temperature perpendicular to the surface.
This relationship is now known as Fourier's law in his honor. However, in his 1807 manuscript
Fourier formulated thermal conductivity mathematically rather than experimentally. As pointed out
by Grattan-Guinness (1972), Fourier arrived at this concept gradually, as he was making ihe tr<m.~tion
from discontinuous bodies to a continuous body. The concept of specific heat capacity, proposed
experimentally by Lavoisier and Laplace in 1783, is an essential part of the transient heat diffusion
process. It helps convert the rate at which heat is accumulating in an elemental volume to an
equivalent change in temperature. Thennal conductivity and thermal capacity are two fundamentally
different attributes of a solid, one governing transport in space and the other governing change in
storage in the vicinity of a point. Together, these two parameters govern the ability of the_ solid. to
respond in time to forces that cause the thermal state of the solid to change. Sometimes, it is found
mathematically convenient to combine the two parameters ·into a single parameter known as thermal
di:ffusivity, 11 = K/(pc). The higher the diffusivity, the faster the tendency of the material to respond
to externally imposed perturbations.
INFLUENCE AND CONNECTIONS
Soon after the publication of the Analytic Theory of Heat in 1822, the general scientific
~onnnunity became aware of the significance of Fourier's work, not merely for the science of heat,
but in general as a rational framework for conceptualization for other branches of science. Within
a few years, the heat conduction analogy was brought to the study of electricity and later to the
analysis of molecular diffusion in liquids and solids. The dynamical theory of gases directly led to the
analogy between diffusion of gases and diffusion of heat. The investigation of the flow of blood
through capillary veins and the flow of water through porous materials led to the adoption of
Fourier's heat_ conduction model to the flow of fluids in geologic media. The study of random
motions of particles led to the interpretation of Fourier's equation in terms of stochastic differential
equations.
Page 12
Simultaneously, Fourier's work began also to be recognized by the establishments of the
intellectual world (Grattan-Guinness, 1972). He was made a foreign member of the Royal Society
in 1823, and in 1827 he was elected to the Academie Fran~aise and the Academie de Medicine. He
succeeded Laplace as the president of the Council of Prefects of Ecole Polytechnique. He also
became the Secretaire Perpetuel of the Academie des Sciences.
For the sake of completeness, it is pertinent here to allude to the fate of Fourier's political
career. In 1815, with the fall of Napoleon at Waterloo, Fourier's political career came to an end. His
pension was refused and, close to fifty years old, he was virtually without an income. But, thanks
to a fonner student of his at the Ecole Polytechnique in 1794 who was a prefect of the department
of Seine, Fourier was given the directorship of the Bureau of Statistics in Paris. Later, in 1817, he
was elected to a vacancy in physics in the Academie des Sciences. With these, Fourier had a secure
income for the rest of his life and he could find plenty of time for conducting research. During the -
1820s Fourier also had an influential and distinguished following: Sturm, Navier, Sophie Germain,
Dirichlet, and Liouville.
To gain an understanding of Fourier's influence over the past nearly two centuries, it is
convenient to organize the discussions into the folloWing subheadings: electricity, molecular diffusion,
flow in porous materials, random walk, and economics.
ELECTRICITY
The nature of electricity and its relation to magnetism were not completely understood at the
time Fourier published his Analytic Theory, nor were the relations between electrostatics and
electrodynamics (galvanic electricity). Becquerel and Barlow, and Davy (Dictionary of Scientific
Biography, vol10, p. 186-194) had been studying the electrical conductibility of metals in the context
of materials with different lengths and cross sectional areas. Quantities such as current strength and
intensity were not precisely defined. At this time, Georg Simon Ohm of Gerinany ( 1789-1854) set
himself the task of removing the ambiguities about galvanic electricity with mathematical rigor,
supported by experimental data. He published four papers on galvanic current between 1825 and
1827, of which the most well-known is his 1827 pamphlet, "Die galvanische Kette, mathematisch
bearbeitet." Ohm's work, which is considered to be one of the most important fundamental
Page 13
contributions to electricity, was largely inspired by Fourier's heat conduction model It was thus a
combination of inductive reasoning as well as an empirical idealization of the phenomenon of
electricity. Ohm started with three "laws" (Ohm, 1827). According to his first law, the
connnunication of electricity from one particle only takes place directly to the particle next to it, so
that no immediate transition from that particle to any other situated at a greater distance occurs.
Recall that Fourier made this important idealization when making the transition from action at a
distance to the continuous medium The second law was that of Coulomb relating to the effect of
a charge at a distance in a dielectric medium The third law was that when dissimilar bodies touch one
another, they constantly maintain the same difference of potential at the surface of contact. This
assumption is quite important because it points to a significant difference between the processes of
heat conduction and conduction of electricity. In the case of heat conduction, temperature is
continuous at material interfaces, whereas in the case of galvanic electricity the potential, namely,
voltage, is discontinuous, as implied by this assumption of Ohm.
By means of carefully controlled experiments, Ohm showed that the current in a galvanic
circuit did not vary with time (steady flow), the intensity of the electric currrent was directly
proportional to the drop in voltage along the conductor in the direction of flow and inversely
proportional to the resistance of the conductor. In turn, the resistance of the conductor was a
function of the material of which the conductor is made of and its form (that is, shape and size).
Equally important, Ohm showed that the resistance of the conductor was independent of the
magnitude of the current itself or the magnitude of the voltage drop (the electromotive force), or the
absolute value of the potential at which the conductor is maintained. In addition to giving precise
meaning to current, electromotive force and resistance, Ohm's work provided a link between
electrostatics (from which the notion of a potential or voltage is derived) and electrodynamics or
galvanic electricity. Following Ohm's work, the measurement of the electrical resistance of various
materials with great precision became a fundamental task in physics (Maxwell, 1881).
In his classic 1827 work Ohm took the analogy with heat conduction much further. In fact,
he treated the flow of electricity as being exactly analogous to the flow of heat and wrote a transient
equation of the form similar to (1)1,
1 Unless otherwise stated, the notations used in this paper are those of the referenc;ed authors.
Page 14
du d 2 u be (3) Y dt = x-- - - u
dx 2 w ·
where y is a quantity analogous to heat capacity, which, according to Ohm, was not experimentally
proven, u is the electric potential (voltage), x is electrical conductivity, b is a transfer coefficient
associated with the atmosphere to which electricity is being lost by the conductor according to .
Coulomb's Law , c is the circumference of the conductor, and w is the area of cross section of the
conductor along the x direction. Ohm was not confident about this equation and admitted that no
experimental evidence for y was as yet forthcoming.
Maxwell (1881) derived the same equation in a different context and showed that Ohm was
in error in proposing (3) the way he did. Maxwell considered a long conducting wire (such as a
transoceanic telegraph cable) surrounded by an insulator. In this case, the insulator, which is a
dielectric material, functions as a condenser and possesses the electrical capacit~c~- _prop~rty
analogous to heat capacitance. Moreover, if the insulator is not perfect, some amount of electricity
would be lost to the surroundings, as indicated by the second term on the right-hand side of (3).
Maxwell expressed Ohm's error thus (Maxwell, 1881, p. 422): "Ohm, misled by the analogy between
electricity and heat, entertained an opinion that a body when raised to a high potential becomes
electrified throughout its substance, as if electricity were compressed into it, and was thus by means
of an erroneous opinion led to employ the equations off.ourier to express the true laws of conduction
of electricity through a long wire, long before the real reason of the appropriateness of these
equations had been suspected." Indeed it is fundamental to the nature of electricity that capacitance
is an electrostatic phenomenon and only insulators possess that property. Electricity, as Maxwell
pointed out (Maxwell, 1881, p. 336), behaves like an incompressible fluid, and hence conductors do
not possess the property of capacitance.
It is of interest to take note of the particular way Ohm formulated his mathematical ideas.
Fourier's law of heat conduction is expressed as: heat flux in the x direction= -K(dT/dx)A, where
A is the area of cross section through which heat is flowing. That is, heat flux is expressed in such
a way that the material property, K, is kept distinct from the geometric attributes of cross sectional
area and distance. However, Ohm expressed current as being equal to the difference in voltage
Page 15
divided by resistance. The resistance in Ohm's law is an integral which combines the material
property as well as the geometry of the conductor of finite size through which current is flowing.
Ohm conceptualized in terms of potential difference whereas Fourier conceptualized in terms of
potential gradient. Fourier's method of separating material property from geometry was of the right
mathematical form to pose the problem as a differential equation. As we shall see later on, while
discussing Fick's work on molecular diffusion, Ohm's approach of dealing with resistance may prove
to be quite advantageous for problems in which the domain of flow is characterized by curvilinear
flow paths and the domain lacks simple symmetry.
Ohm's work is now accepted as one of the most important contributions in the science of
electricity. Yet, recognition did not come to him readily. Although physicists such as Theodor
Fechner, Heinrich Lenz, Wilhelm Weber, Friederich Gauss, and Moritz Jacobi drew upon Ohm's
work in their own research soon after Ohm published "Die galvanische Kette", Ohm's work came
under criticism from an unexpected quarter. His experimental approach to finding order in nature
was heavily criticized by Georg Poul (Dictionary of Scientific Biography, 1981 ), a physicist who was
a follower of Hegel's philosophy of pure reason. However, due recognition came to Ohm after a few
years when he was elected to the Academies at Berlin and Munich and the Royal Society conferred
on him the Copley Medal in 1841. Ohm moved to Cologne and went on to occupy the Chair of
Physics at that University.
William Thompson ( 1824-1907) was greatly influenced by Fourier's work even when he was
in his teens. Thomson's first two articles, written at ages 16 and 17, were in defense of Fourier's
mathematical approach. Later he demonstrated the similarities between the mathematical structures
of Fourier's heat conduction equation and the equations of electrostatics stemming from the works
of Laplace and Poisson (Thompson, 1842). For example, potential was analogous to temperature,
a tube of induction was analogous to a tube of heat flow, the electromotive force was in the direction
of the gradient of potential and the flux of heat was in the direction of temperature gradient.
While physical analogies serve a useful purpose, Maxwell (1888, pp. 52-53) emphasized that
caution was in order so that the analogies are not carried too far. Maxwell pointed out that the
analo~ with e~ectric phenorrena applied only to the steady flow of heat. Even here differences exist
between electricity and heat. For steady flow, heat must be kept up by a continuous supply,
Page 16
accompanied by its continuous loss. However, in electrostatics, a set of electrified bodies placed in
a perfectly insulating medium might remain electrified forever without any supply from external
sources. And, there is nothing in the electrostatic system that can be described as flow. Another
limitation of the analogy is that the temperature of a body cannot be altered without altering the
physical state of the body, such as density, conductivity, or electrical properties. However, the
electrical potential is merely a scientific concept. Bodies may be very strongly electrified without
undergoing any physical change.
We saw earlier that Ohm had attempted unsuccessfully to formulate a time-dependent
electrical flow problem by direct analogy with Fourier's equation. Later work, stemming from
Maxwell's equations, established that transient heat conduction and transient electricity flow are very
different in nature. Transient flow of electricity typically arises in the case of alternating current as
opposed to the steady state direct current with which Ohm was concerned. In the case of alternating
current, the change in electric field is intrinsically coupled with an induced magnetic field in a
direction perpendicular to the direction in which current is flowing. The nature of the coupled
phenomena is such that when the frequency of the alternating current is low, Maxwell's
electromagnetic equations may be described in the form of an equation which looks mathematically
similar to the heat conduction equation, in that one side of the equation involves the Laplace operator
(second derivative in space) and the other involves the first derivative in time. However, the
resemblance is only superficial because the dependent variable in this.equation is a vector potential,
whereas the dependent variable in the heat conduction equation is a scalar potential.
MOLECULAR DIFFUSION
Molecular diffusion is the process by which molecules of matter migrate within solids, liquids
and gases. The phenomenon of diffusion was observationally known to chemists and biologists by
the eighteenth century. In the early nineteenth century, experimental chemists began paying serious
attention to molecular diffusion, and the publication of Fourier's Analytical Theory in 1822 provided
the chemists with a logical framework with which to interpret and extend their experimental work.
The following discussion on molecular diffusion starts with diffusion in liquids, followed by solids and
gases.
Page 17
Diffusion in Liquids
Among the earliest observations which attracted the attention of chemists to diffusion in
liquids is the phenomenon of osmosis. In 1752, Jean Antoine (Abbe) Nollet (1700-1770) observed
and reported selective movement of liquids across an animal bladder (semipermeable membrane).
Between 1825 and 1827, Joachim Henri Rene Dutrochet (1776-1847) made pioneering contributions
in the systematic study of osmosis. A physiologist and medical doctor by training, Dutrochet spent
most of his career in the study of the physiology of animals and plants. About this time~ Poisson had
attempted to explain osmosis in terms of capillary theory. In his paper, Dutrochet (1827) strongly
disagreed with Poisson and, based on experimental evidence, argued that two currents (solute and
solvent) simultaneously occur in opposite directions during osmosis, one of them being stronger than
the other, and that the understanding of osmosis required something more than a simple physical
mechanism such as capillarity. He speculated on the possible role of electricity in the osmotic
phenomenon. He also coined the terms "endosmosis" for the migration of the solve~t!owards the
solution and the term "exosmosis" for the reverse process.
The next major work on liquid diffusion was that ofThomas Graham (1805-1869). Graham's
experimental work on liquid diffusion led to the distinction between crystalloids (such as common
salt) having high diffusibility and colloids (such as gum arabic) having low diffusibility. Graham's
detailed observations on the diffusibility of a variety of chemical substances would later give him the
distinction of being considered by some to be the father of colloid chemistry. In 1850 he presented
data on the diffusibility of a variety of solutes and solvents in his Bakerian Lecture of the Royal
Society.
Despite the wealth of data he collected, Graham did not attempt to elicit from them a unifying
fundamental statement of the process of diffusion in liquids. That Graham restricted himself
essentially to the collection of experimental data on diffusion in liquids proved to be a catalyst for one
of the most influential papers of molecular diffusion, that of Fick (1855). AdolfFick (1829-1901)
was a Demonstrator in Anatomy at the University of Zurich and, despite his professional training in
medicine, had a sound background in mathematics and physics. Barely twenty five years old, Fick
started by expressing regret that Graham had failed to identify any fundamental law of diffusion from
his substantial experimental data and set himself the task of remedying the situation. Fick saw a direct
Page 18
analogy between the diffusion of heat in solids and the diffusion of solutes in liquids, and his starting
point was Fourier's heat conduction equation.
By direct analogy with Fourier, Fick wrote down the parabolic equation for transient diffusion
of solutes in liquids in one dimension thus:
(4) o(~ + l. dQ~) ox 2 Q dx ox
oc --ot
where D is th~ diffusion coefficient, c is aqueous concentration, and Q is the area of cross section.
It is of particular interest to note that Fick made a novel departure from Fourier in writing the one
dimensional equation. Note that the second term on the left hand side of (4) accounts for the
variation of the area of cross section along the flow path (the x axis). Intrinsically, therefore, Fick's
equation is valid for a flow tube of arbitrary shape involving a curvilinear x axis. Indeed, Fick (1855)
presented data from a diffusion experiment in an inverted funnel shaped vessel, solved (4) for the
particular cone-shaped vessel and found that his mathematical solution compared favorably with the
steady state observations made at different locations within the vessel For a flow tube with constant
area of cross section, (4) simplifies to Fourier's equation, and one can readily verify that (4) leads also
to the appropriate differential equations for radial and spherical coordinates. Upon reflection, it
becomes apparent that integration of ( 4) along curvilinear flow tubes leads to the evaluation of
resistances within finite segments of flow tubes and that the evaluation of resistances thus . provides
a link between the approaches of Fick and Ohm.
According to Fick, concentration is analogous to temperature, heat flux is analogous to solute
flux and thennal diffusivity is analogous to chemical diffusivity. If concentration in the aqueous phase
is defined as mass per unit volmre, then specific chemical capacity (analogous to specific heat) equals
unity and chemical diffusivity is equal to chemical conductivity.
In the second part of his paper, Fick (1855) went on to analyze flow across a semipermeable
membrane by idealizing the membrane as a collection of cylindrical pores of radius p. As suggested
Page 19
earlier by Dutrochet (1827), two simultaneous currents will occur through the capillaries; the solute
current will occur towards the solvent and the solvent current will occur towards the solution. Fick
reasoned that because of the affinity of water to the material comprising the membrane, the water
current will be organized more toward the walls of the pores and the solute towards the axis of the
pores. Incidentally, a remarkably similar reasoning was employed by Taylor (1953), who studied
solute diffusion in capillary tubes with moving water. When the radius of the pore becomes
sufficiently small, the flow of the solute will be arrested and osmosis will involve one current, that of
the solvent. Interestingly, Flck did not address the issue of the forces causing the movement of water,
that is, why the solute should move in the direction of concentration gradient. He simply asserted that
in a diffusion vessel, as the solute moves one way, a certain amount of water will move in the
opposite direction. Fick went on to publish a year later what probably is the first text book in
biophysics under the title, Die medizinische Physik (1856).
The study of liquid diffusion was soon to take a very important place in the field of biophysics
through the investigations of Wilhelm Pfeffer (1845-1920). After receiving a doctoral degree in
chemistry from Gottingen when he was twenty years old, Pfeffer grew interested in the study of
biological processes and brought his experimental and analytical skills to bear on the study of mass
transfer in plant cells. Broadly, the outer layer of the cell was treated as a semipermeable membrane,
and Pfeffer devised sophisticated techniques to measure osmotic pressure within cells, and went on
to develop and test several hypotheses concerning the diffusion of nutrients within and across cells.
Pfeffer found osmosis experiments on plant cells to be quite limiting and sought to conduct
measurements on controlled inorganic membranes. Along these lines he pioneered the use of thin
layers of ferrocyanide deposited on ceramic substrates as semipermeable membranes. Using such
membranes he went on to measure osmotic pressure of various solutions as a function of
concentration as well as temperature. Pfeffer's data, published in his 1877 classic, Osmotische
Untersuchung_en, would later help van't Hoff to lend credibility to his theory of osmotic pressure.
Dutrochet, Pfeffer, Fick and other biophysicists of the time strongly supported the view that
physiological processes must be elucidated and understood in terms of inorganic (nonbiological)
processes. Dutrochet (1827) eloquently articulated this view thus: "The physical processes of living
and inorganic matter merge in endosmosis and exosmosis. The further we advance in our
Page 20
understanding of physiology, the more reasons we will have to revise our opinion--whose major
proponent is Monsieur Bichat--that life and physical phenomena are essentially different. It is
undoubtedly false."
By the time Preffer published his book on cell mechanics, a wealth of data had been collected
on osmosis, both from physiological and inorganic materials. Many hypotheses were in vogue and
a rational desciiption of osmosis in terms of known principles of physics and chemistry was lacking.
Jacobus Henricus van't Hoff (1852-1911), one of the most influential physical chemists of the second
half of the nineteenth century, filled this gap by providing a theoretical foundation for osmotic
pressure based on well-established laws of chemistry. Van't Hoff (1887) started with and justified
the proposition that the physical behavior of solutions and the associated osmotic pressure can be
rationally understood by treating solutions as analogous to gases and by applying Boyle's law, Gay
Lussac's law, and Avogadro's law to solutions. He formally defined osmotic pressure as the excess
pressure that would develop in a solution contained in a vessel that conununicates with a reservoir
of a solvent across a perfect semipermeable membrane. By using the aforesaid laws and the second
law of thermodynamics, van 't Hoff was able to draw many inferences about relationships between
the magnitude of osmotic pressure on the one hand and the nature of the solute concentration and
temperature on the other. He demonstrated that the experimental data of previous workers,
especially Pfeffer (1877), substantially justified his theoretical framework.
In osmosis, as we have seen, two opposing currents of flow are involved and each current is
driven by its own force--the solvent by spatial variations in its fluid potential and the solute by the
spatial variations of osmotic pressure. Therefore, it is convenient to conceptualize the total pressure
in the solution as a sum of the water phase pressure and the osmotic pressure. Thus, in the solution,
the pressure in the water phase Pw = Ptota~ - Posmotic. The stronger the concentration of the solution, the
lesser is the water phase pressure and the stronger will be the solvent current towards the solution
should the solution communicate with the solvent across a semipermeable membrane. Analogously,
the solute will be driven in the opposite direction because osmotic pressure decreases in the direction
of the solvent.
Closely following van't Hoff, Walther Hermann Nemst (1864-1941) examined the process
of solute diffusion in the context of osmotic pressure as defined by the former. Nemst ( 1888) pointed
Page21
out that the diffusion of solutes in the direction of decreasing concentration had been suggested
earlier by Berthollet (1803) and that Fick established it rigorously with mathematics, supported by
experirrental data. Nernst found Fick's approach to be formal and lacking in the elucidation of the
forces which impelled the solute diffusion process. To overcome this deficiency, he looked at
diffusion in terms of impelling forces and resistive forces, the former stemming from spatial variations
of osmotic pressure and the latter stemming from the collision of molecules with the solvent
molecules and even among the solute molecules themselves.
Nernst (1888) considered the force due to osmotic pressure acting on a molecule of the solute
and defined a _coefficient of resistance K representing the force required to move 1 gram-molecule
of the solute through the solvent at a velocity of 1 em/sec. Combining these, he expressed the flux
of solute in terms of the gradient of the osmotic pressure and the reciprocal of the resistance
coefficient K. He then recognized that for dilute solutions, osmotic pressure is linearly related to
concentration by a simple relation, p = p0c, where Po is the osmotic pressure in a solution containing
a gram molecular weight of the solute and c is concentration. As a result, for dilute solutions, the
ratio pc/K becomes part of the diffusion coefficient and flux becomes proportional to the gradient of
concentration, as proposed by Fick (1855). By extending the analysis to concentrated solutions,
Nernst pointed out that in such solutions the solute will encounter greater resistance to flow because
of mutual collision among the solute molecules in addition to the solvent molecules. Therefore, in
concentrated solutions the diffusion coefficient will be a function of concentration. As a
consequence, the relevant differential equation of diffusion becomes non linear.
In analyzing the process of diffusion, Nernst gave consideration to electrolytes in which
individual ions will migrate separately. In this case, in order that the ions composing a given solute
may migrate at the same velocity, he suggested that the differences in ion velocities induced by
osmotic pressure will be compensated by electrostatic forces.
Solid Diffusion
An early documented observation of solid diffusion is attributed to Robert Boyle (Barr, 1997)
who su~c~eded in 1684 in making zinc diffuse into one of the faces of a copper farthing, leading to
the formation of brass. By carefully filing the face, Boyle showed that zinc had indeed diffused into
Page 22
the body of copper. Yet, controlled diffusion measurements would not become possible until two
hundred years later.
The first rreasurements of the diffusion of one solid metal into another was made by William
Roberts-Austen (1843-1902) who was Chemist and Assayer of the British Mint. He started his career
at the Mint as an assistant to Thomas Graham, who was the Master of the Mint from 1855 to1869,
and extended the scope of his diffusion studies to metals and alloys. Roberts-Austen thus took up
the challenge of extending Graham's work on liquid diffusion to metals. However, his progress was
considerably hampered because of difficulties in accurately measuring the temperature at which
diffusion was taking place in the solid state. Finally, he succeeded by adopting Le Chatelier's
platinum-based thermocouples and was able to study the diffusion of gold in solid lead at different
temperatures. The results were analyzed in terms of Fourier's model of one-dimensional diffusion
(Roberts-Austen, 1896).
The discussion of solid diffusion would not be complete without a mention of the work of
Enrico Fermi (1901-1954). He was the first to successfully achieve, in 1942, a sustained release of
energy from a source other than the sun by bombarding and splitting uranium atoms with the help of
neutrons slowed down in a matrix of solid graphite. Critical to the design of the experiment was the
calculation of the slowing down of neutrons and the absorption of thermal neutrons by the carbon
host. The slowing down of the neutrons was described as a diffusion process (Anderson and Fermi,
1940) and the corresponding diffusion constants were calculated based on experimental data . The
approach was one Fermi had already perfected earlier (Fermi, 1936). The diffusion theory developed
by Fermi would later be known as the "age theory."
Diffusion of Gases
The earliest experimental work on the diffusion of gases was by Graham (1833). When two
or more gases are mixed together in a closed vessel, the natural tendency is for the gases to
redistribute themselves by diffusion in such a way that the mixture has a uniform composition
everywhere. Graham showed experimentally that the rate at which each of the gases diffuses is
inversely proportional to the square root of its density. This observation was stated as a Jaw by
Graham (1833) and is known as Graham's law. When we compare gas diffusion with liquid diffusion
Page 23
or heat conduction or the conduction of electricity, we find that in these latter cases we are concerned
with conductive transport in different materials, whereas in the case of gas diffusion we are concerned
with the conduction of gas in free space. In the case of nongaseous conduction, the transport
coefficient (conductivity or diffusivity) is experimentally estimated for different materials on the basis
of Fourier's equation. The conductivity of different materials pertains to the ability of the materials
to inhibit or resist the flow of the permeant. In contrast, in the case of pure gaseous diffusion,
diffusivity is a property that stems solely from the attributes of the permeating fluid, the gas.
With the advances that were taking place in molecular physics and chemistry during the
middle of the nineteenth century, a great deal of effort ·was being made by researchers to directly
estimate the properties of gases such as viscosity, specific heat, thermal conductivity, Diffusion
coefficient an~ diffusivity by starting with force, momentum and energy at the molecular level and
statistically integrating these quantities in space and time to estimate the macroscopic properties of
interest. Among the earliest researchers in this regard was Maxwell (1831-1879), whose work on
the dynamical theory of gases is of fundamental importance. In this work, Maxwell ( 1867) assumed
molecules to be small bodies or groups of small bodies which possess forces of mutual repulsion
varying inversely as the fifth power of distance. Macroscopically, he described the diffusion of a
mixture containing two gases in terms of an equation which has the same form as Fourier's transient
heat conduction equation. In this case, the diffusion coefficient is describable by Dalton's Law of
partial pressures and densities of the two gases and is inversely proportional to the total pressure.
Maxwell generated a solution of this equation for the case of a particular column experiment
conducted by Graham involving carbonic acid and air and found some agreement with the diffusion
coefficient independently estimated by Graham.
Brownian Motion
About the same time osmosis was being recognized as an important physical process by
Dutrochet, Robert Brown (1773-1858), a renowned British Botanist discovered that pollen and other
fine particles suspended in water exhibited continuous and permanent random motions. During the
second half of the nineteenth century, physicists became very interested in the mechanisms and forces
responsible for these continuous motions. It was soon recognized through the investigations of
Page 24
Delsaux, Gouy and others (Fiirth, 1926) that the random motions were sustained by the impacts of
the molecules of the liquid on the suspended by particles. Subsequently, the theory of Brownian
moverrent was placed on formal physical-mathematical foundations by Albert Einstein (1879-1955)
in one of his celebrated papers (Einstein, 1905). In this paper Einstein set out to establish the validity
of the molecular-kinetic theory of heat rather than attempting to explain Brownian motion. His goal
was to show· that because of the molecular-kinetic nature of heat, bodies of microscopically
observable particles suspended in liquids will perform movements of such magnitude as are
observable under a microscope.
Einstein started with the proposition that colloidal particles suspended in liquids exert osmotic
pressure, just as dissolved solute molecules and that an equal number of suspended colloidal particles
and nonelectrolyte solute molecules exert the same osmotic pressure in dilute solutions. Such
osmotic pressure arises from the random motion of the particles as they are impelled by their random
collisions with the vibrating liquid molecules. The kinetic energy transferred in the process is directly
related to the temperature of the liquid (analogous to Nemst's idealization of osmotic pressure in
solutes). In their random movement, the particles are decelerated by the viscous resistive forces of
the liquid. By analogy with van't Hoff's equation for osmotic pressure for none1ectrolytic solutes,
the osmotic pressure associated with suspended particles is given by,
RT (5) p = -v
N
where pis the osmotic pressure, R is the universal gas constant, Tis temperature, N is Avogadro's
Number, and vis the number of suspended particles per unit volume of the liquid. Also, the resistive
force offered by the particles suspended in a unit volume of the liquid is given, according to Stokes
formula by,
(6) vK = v61tkPu,
Page 25
where Kis the resistive force per particle of radius P, k is viscosity and u is particle velocity. Noting
that the particles are macroscopically impelled by spatial gradient of osmotic pressure and that under
dynamic equilibrium. impelling forces and resistive forces must balance, Einstein derived a
macroscopic fiux law (analogous to Fourier's law for thermal conduction) for the flux of particles
crossing a unit area in unit time,
(7) D ov =
ax vK
61tkP ,
where D, the diffusion coefficient is given by D = RT/{61tkPN), in which N is Avogadro's Number.
In view of (7), it would have been trivial for Einstein to have derived a partial differential
equation for colloid diffusion in a manner analogous to Pick's derivation. However, Einstein had a
far more fundamental goal and followed a different path. He proceeded to derive the partial
differential equation for the distribution of particles at time t + 1:, given that the distribution at time
tis v = f (x,t). During a time interval 1: there exists a finite probability <J>(A)da that the x coordinate
of a single particle will change by an amount A. This leads to the recursive relation,
(8)
f ( x, t + 1:) = J f (x +A, t) <P (A) d A .
For small values of 't, the left-hand side of (8) can be approximated by f(x,t+'t) z f(x,t) + 1: {af/ot).
Also, f(x+A,t) can be approximately expanded into a Taylor's series,
(9) ar A2 a2 r f (x + A , t) = f (x , t) + A- + - --
ax 2 a x 2
Einstein assumed the function <I> to be such that <P (A)= <P (-A) and that <1> differs from zero only for
small values of A. Under such assumptions, fA <1> (A)= 0 and f A2 <I> (A)= 1. Thus, substitution of
(9) into (8) leads to the one-dimensional diffusion equatiOJ?.,
Page 26
(10) af = 0 a2f ,
at ax 2
where D = ( 1/'t) f _ .. (a 2/2) <P (a) ~ . The distribution function f(x,t), which is a solution to the
diffusion equation (10), is a symmetric function with a mean value of zero.
Einstein extended f (x,t) to represent the probability that n particles, each with its own
COOrdinate system, will have displacements between X + ax after time t and showed that the function
can be expressed as,
x2 --(11) f(x,t) =
~4rtD
n e 4Dt
{t
He recognized the similarity between this function and that representing the distribution of random
error by stating, ''The probability distribution of the resulting displacements in a given time t is
therefore the same as that of fortuitous error, which was to be expected." We will see later that this
line of reasoning had been used by Edgeworth in 1883 to derive the diffusion equation relating to the
Law of Error.
Einstein's 1905 paper is considered a landmark in physics. It provided a strong impetus to
experimentally establish the veracity of the molecular-kinetic theory of Brownian motion as well as
to detennine Avogadro's Number more precisely.
Soon thereafter, Paul Langevin (1872-1946) developed an alternate approach to describing
Brownian movement. Rather than devoting attention to the distribution of particles in space (as
Einstein did), Langevin (1908) started with the forces acting on a single particle. A Brownian particle
is impelled by the momentum transferred to it by the molecules that collide with it. In tum, the
particle is retarded by the viscous resistance offered to it by the molecules of the liquid. Thus the net
force on the particle equals the sum of a "systematic" drag force and a stochastic force (Langevin's
complementary force),
Page 27
(12) F ( t ) = -61t p au + X ( t ) ,
where p is viscosity, a is the radius of the spherical particle, u is velocity, and X is the stochastic
force. Langevin showed that after a "relaxation" tirre 't of the order ofm'(61tpa), where m is the mass
of the particle, the diffusion equation of Einstein (1905) is valid. Langevin's equation soon became,
among physicists, the starting point for formulating stochastic differential equations. In particular,
Fokker, Planck and others showed that Langevin's equation can be used to represent the diffusion
process in terms of the probability W (u+Llu) that the velocity of a particle is in the interval u and
u+Llu at the end of an interval of time -r, given that it had a velocity u at time zero. This has led to
the equation describing diffusion in the velocity space, known as the Fokker-Planck equation
(Chandrasekhar, 1943, p. 33),
(13)
where P = m'(61tpa) and q = Pkt/m, in which k is Holtzman's constant, tis absolute temperature and
m is the mass of the particle. In (13) the first term on the right-hand side is known as "drift." As we
shall see later, Bacheller (1900) had independently developed this type of stochastic differential
equation in developing the theory of speculation in connection with the pricing of options in the
French stock market.
The Fokker-Planck equation is considered to be a more general representation of the
stochastic process of diffusion than Einstein's equation. The basis for this consideration is the notion
of a Markoff Process, which is a random process without "memory". In applying Langevin's
equation to Brownian movement, it is assumed that velocity is assumed to be a Markoff process.
Note that the position x of the Brownian particle is an integral of velocity. Strictly speaking, if
velocity is a Markoff process, position will have some memory and so it cannot be a Markoff process
(Gillespie, 1996). Therefore, to enable stochastic analysis of position, the derivation of the Fokker
Planck equation has to start with a multivariate form of the Langevin equation in which velocity and
position together constitute a stochastic process. In this framework, Einstein's equation arises when
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the time interval chosen is suffiCiently large. If, however, the time interval is small, the mean value
of the probability distribution will "drift" away from zero and is accounted for by the first term on the
right hand-side of (13). Additionally, drift could also arise from the presence of external forces such
as gravity. The equation is then referred to as Smoluchowski's equation.
FLOW OF WATER IN POROUS MATERIALS
Fourier's heat conduction equation has had an enormous influence in the study of fluid flow
processes in the earth, especially water and petroleum in porous media. In applying the equation to
these processes, the following analogies can be made: temperature corresponds to scalar fluid
potential, heat corresponds to mass of water, thermal conductivity corresponds to hydraulic
conductivity and heat capacity corresponds to hydraulic capacity. However, unlike electricity, heat,
and solutes, the potential of water has a very special attribute, namely, gravity. This attribute renders
the extension of heat analogy to the earth sciences particularly interesting.
Steady Flow of Water
By the tirre of Fourier's work, fluid rrechanics was a well developed science and the concept
of a fluid potential, defined as energy per unit mass of water was already established through the
seminal contribution of Daniel Bernoulli in hydrodynamics, early in the eighteenth century. The flow
of water in open channels was being rigorously studied by civil· engineers. In addition to civil
engineers, many physiologists were also interested in the study of water flow through capillary tubes
to better understand, by analogy, the flow of blood through narrow vessels.
Among the earliest experirrentalists to study the slow motion of water through capillary tubes
was Jean Leonard Marie Poiseuille (1799-1869), a physician and a physiologist who had studied
under Ampere at the Ecole Polytechnique. He was not satisfied with the contemporary state of
understanding of blood circulation in veins and therefore embarked on a study of the flow of water
in narrow capillary tubes under carefully controlled conditions. Using a sophisticated laboratory
setup, Poiseuille studied the flow of water in horizontal capillary tubes varying in diameter from
about 50 to about 150 microns and measured fluxes as low as 0.1 cc over several hours. In the
absence of gr~vity, he found that water flux was directly proportional to the pressure difference
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between the inlet and the outlet and inversely proportional to the length of the capillary. These
observations were very similar to those made by Ohm in the case of galvanic current. Although the
work was completed in 1842, it was not published until a few years later (Poiseuille, 1846). Similar
observations had been made earlier in Germany by Hagen (1839)
One of the most influential works on the flow of water in porous media during the nineteenth
century was that of Darcy (1856). Henry Darcy (1803-1858) was a highly recognized civil engineer
who is credited with designing and completing the first ever protected town water supply system in
the world. Dissatisfied with the unhealthy sources of drinking water available in his native town of
Dijon, he helped bring and distribute water from a perennial spring located several kilometers away
from the town..The project was completed in 1840. Later, presumably to build a water purification
system, Darcy conducted a series of experiments in vertical sand columns to develop a quantitative
relationship for estimating the rate of flow of water through sand filters. Darcy's experiment was
novel in that it included gravity and it involved a natural material (sand) rather than an engineered
material such as a capillary tube. He too, like Ohm and Poiseuille before him, found that the flux of
water through the column was directly proportional to the drop in potentiometric head, h = z + lJ1,
where z is elevation with reference to datum and lJ1 is pressure head, directly proportional to the area
of cross section and inversely proportional to the length of the column. Darcy's law plays a
fundamental role in many branches of earth sciences such as hydrogeology, geophysics, petroleum
engineering, soil science, and geotechnical engineering.
The middle of the nineteenth century witnessed many developments in the earth sciences,
among which was the discovery of artesian groundwater basins in France and elsewhere in Europe.
It was apparent that the deep circulation of groundwater driven by gravity could be rationally
understood within the context of potential theory. Soon, the heat conduction model of Fourier began
to be used for analyzing circulation of water in groundwater basins. The earliest studies in this regard
restricted themselves to the steady motion of groundwater. Unlike the problems of electricity and
molecular diffusion, the problems of groundwater involved large spatial scales (many tens or even
hundreds of kilometers laterally and hundreds of meters vertically). Two of the most distinguished
engineers of this era were Jules-Juvenal Dupuit (1804-1866) in France and Philipp Forchheimer
(1852-1933) in Austria. Dupuit (1863) developed the basic theoretical framework for analysis of
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flow in groundwater systems and of the flow of water to wells. Forchheimer (1886) formally stated
the steady seepage of water in tenns of the Laplace equation and initiated the use of complex variable
theory to the solution of two-dimensional problems in flow domains of complicated geometry that
occur in the vicinity of dams and other engineering structures. He also pioneered the use of flow nets
as practical graphical means of solving seepage problems in complex flow domains.
Flow _of Multiple Fluid Phases
A significant development in the study of flow in porous media was the work of Edgar
Buckingham (1867-1940), a distinguished thermodynamicist of his time. From 1902-1905 he was
an Assistant Physicist with the Bureau of Soils, U.S. Department of Agriculture. In this brief period,
he not only introduced himself to a totally new field, the science of soils, but made one of the most
important contributions to soil physics in particular as well as to the general study of multiphase fluid
flow. Soon after, he moved to the National Bureau of Standards and became well known for, among
other achievements, for his 1t-theorem of dimensional analysis.
In soils close to the land surface where plant roots thrive, both water and air coexist. An
understanding of the dynamics of the occurrence and movement of water in the soil is critical to
agricultural managemmt When water and air coexist in soils, the contacts between air and water in
the minute pores are curved menisci in which energy is stored. As a result, the pressure in the water
phase is less than that in the air phase and the difference is the capillary pressure. The physics and the
mathematics of capillarity had been enunciated a hundred years earlier by Laplace and by Thomas
Young. Buckingham brought together the work on capillary pressure with that of Fourier and Ohm
on diffusion, and defined the capillary potential in the water phase as a sum of work to be done per
unit mass against gravity and fluid pressure. He stated that moisture moves in soils in response to
spatial variations in potential and that moisture flux density is directly proportional to the gradient of
capillary potential, the proportionality constant being hydraulic conductivity. Although this statement
resembles the laws of Fourier, Ohm, and Darcy, there exists a very important difference. In soils
which contain water and air, the capillary potential is directly related to water saturation. And, as
water saturation decreases, the flow paths available for mositure movement decrease. Therefore, the
conductivity parameter is strongly dependent on capillary potential, instead of being constant or
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nearly so as is the case with the laws of Fourier, Ohm, or Darcy. Because Buckingham rigorously
defined the capillary potential, his flux law encompasses Darcy's law as a special case of full water
saturation. The strong dependence of hydraulic conductivity on capillary potential renders the study
of moisture diffusion in soils a very difficult mathematical problem For the first time, Buckingham
also experirrentally treasured the relation between capillary potential and water saturation in different
soils. In an earlier work, Buckingham (1904) also applied the diffusion equation to the migration of
. gas in soils and analyzed the dynamic vertical migration of air from the land surface to the water table
in response to fluctuating atmospheric pressure. Buckingham's work helped resolve a contemporary
paradox in agriculture. In arid regions where evaporation rates are very high, the soils are found to
be wetter and hold their moisture for much longer periods than do the soils of humid areas in dry
seasons. Part of the reason for this counter- intuitive observation is to be found in the dependence
of hydraulic conductivity on capillary potential, or, equivalently, water saturation. In arid areas, as
evaporation rapidly desaturates the uppermost soil, the hydraulic conductivity drops practically to
zero and further evaporative loss from deeper zones is virtually eliminated.
Note that specific heat, originally defined and measured by Lavoisier and Laplace (1783), is
an extretrely important physical attribute of materials and occupies an important position in the heat
conduction equation. It plays a crucial role in dictating the rapidity with which a material will
thermally respond to externally imposed perturbations: the smaller the capacitance, the faster the
response. Analogously, in the phenotrenon of fluid flow in porous tredia, hydraulic capacitance plays
a very important role. It so happens that the slope of the variation of water saturation as a function
of capillary potential contributes to the hydraulic capacitance of a soil As a consequence,
Buckingham's work lies at the foundation of the dynamics of multiple fluid phases in porous media.
Although Buckingham theoretically defined a capillary potential, he could only measure it
indirectly in vertical columns in which water moves down solely by gravity. He recognized that new
instrmrents would have to be developed to measure capillary potential under dynamic conditions of
flow. Such an instrument was invented over a decade later by Willard Gardner (1883-1964). This
ingenious instrwrent is called the tensiorreter. The key component of this device is a porous ceramic
cup that is completely saturated with water. Such a porous cup acts like a sernipenneable membrane,
allowing the flow of water from the soil into the cup, but not allowing the flow' of air. The cup is
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connected to a long tube filled with water which is connected to a manometer. The tensiometer is
set into a natural soil and, through exchange of water between the soil and the cup and, fluid pressure
inside of the cup is allowed to attain equilibrium with that in the soil. The equilibrium pressure
represents the capillary potential. The first measurements from this instrument were reported by
Gardner et al. (1922). Subsequent measurements of the relation between water saturation and
capillary potential on many soils have shown that the relation is not unique and is characterized by.
hysteresis. Thus, the hydraulic capacitance of the soil introduces a strong non-linearity into the
differential equation.
Deformable Porous Media
The attribute of hydraulic capacitance of a naturally occurring porous material such as a soil
or a rock arises also for reasons other than the rate of change of saturation with potential. Earth
materials are deformable in response to changes in the stresses which act on the porous skeleton. The
ensuing rate of change of pore volume (which is occupied by water) in response to changes in fluid
potential also contributes to hydraulic capacitance. The measurement of pore volume as a function
of fluid potential was elucidated through the work of Karl Terzaghi (1883-1963), who founded the
discipline of soil mechanics. In presenting his experimental results on the deformation of water
saturated clays, Terzaghi (1925) postulated that in water saturated earth materials volume change is
to be related to the difference between skeletal stress_es and water pressure. Thus, when skeletal
stresses remain unchanged, volume change is directly attributable to the change in fluid pressure or,
equivalently, the fluid potential Extensive experimental work following Terzaghi has shown that